Branching brownian motion with an inhomogeneous breeding potential

J. W. Harris; S. C. Harris

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 793-801
  • ISSN: 0246-0203

Abstract

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This article concerns branching brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β>0. It is known that for p>2 the number of particles blows up almost surely in finite time, while for p=2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, Rt, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of Rt as t→∞. In the case of constant breeding at rate β the linear asymptotic for Rt is long established. Here, we find asymptotic results for Rt in the case p∈(0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p↑2, and a non-trivial limit for lnRt when p=2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.

How to cite

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Harris, J. W., and Harris, S. C.. "Branching brownian motion with an inhomogeneous breeding potential." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 793-801. <http://eudml.org/doc/78044>.

@article{Harris2009,
abstract = {This article concerns branching brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β&gt;0. It is known that for p&gt;2 the number of particles blows up almost surely in finite time, while for p=2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, Rt, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of Rt as t→∞. In the case of constant breeding at rate β the linear asymptotic for Rt is long established. Here, we find asymptotic results for Rt in the case p∈(0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p↑2, and a non-trivial limit for lnRt when p=2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.},
author = {Harris, J. W., Harris, S. C.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {branching brownian motion; additive martingales; spine constructions; branching Brownian motion; population explosions; right-most particle; asymptotics},
language = {eng},
number = {3},
pages = {793-801},
publisher = {Gauthier-Villars},
title = {Branching brownian motion with an inhomogeneous breeding potential},
url = {http://eudml.org/doc/78044},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Harris, J. W.
AU - Harris, S. C.
TI - Branching brownian motion with an inhomogeneous breeding potential
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 793
EP - 801
AB - This article concerns branching brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β&gt;0. It is known that for p&gt;2 the number of particles blows up almost surely in finite time, while for p=2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, Rt, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of Rt as t→∞. In the case of constant breeding at rate β the linear asymptotic for Rt is long established. Here, we find asymptotic results for Rt in the case p∈(0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p↑2, and a non-trivial limit for lnRt when p=2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.
LA - eng
KW - branching brownian motion; additive martingales; spine constructions; branching Brownian motion; population explosions; right-most particle; asymptotics
UR - http://eudml.org/doc/78044
ER -

References

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  10. [10] A. E. Kyprianou. Travelling wave solutions to the K-P-P equation: Alternatives to Simon Harris’ probabilistic analysis. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 53–72. Zbl1042.60057MR2037473
  11. [11] R. Lyons. A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 217–221. K. B. Athreya and P. Jagers (Eds). IMA Vol. Math. Appl. 84. Springer, New York, 1997. Zbl0897.60086MR1601749
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